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Additional Analysis Techniques
for Linear Circuits
Models and Equivalent Circuits for
Analysis and Design
Kevin D. Donohue, University of Kentucky
1
Example - Superposition

Solve for Io in the 3 circuits.
What is the relationship between
these results.
2A
4
6
Io
2
6
2A
4
4
6
9V
6
2
Io
6
9V
Kevin D. Donohue, University of Kentucky
6
2
Io
2
Linearity and Superposition

If a linear circuit has multiple independent sources, then a
voltage or current quantity anywhere in the circuit is the sum of
the quantities produced by the individual sources (i.e. the result
when all other sources are deactivated). This property is called
superposition.

To deactivate a voltage source, set the voltage equal to zero
(equivalent to replacing it with a short circuit).

To deactivate a current source, set the current equal to zero
(equivalent to replacing it with an open circuit).
Kevin D. Donohue, University of Kentucky
3
Examples

Solve for voltages and currents in circuits
containing multiple sources using the
principle of superposition.
Kevin D. Donohue, University of Kentucky
4
Example - Equivalent Circuit

Find the voltage and currents generated in 3
different loads across terminals AB:
 open
circuit
 resistor RL
 short circuit
A
Is
Rth
B
Vs
Rth
A
B
Kevin D. Donohue, University of Kentucky
5
Results - Equivalent Circuit
Current Source Circuit
VAB
Open
RL
IAB
I s Rth
Short
0
Is
Voltage Source Circuit
0
Is
RL Rth
Rth
Is
RL  Rth
RL  Rth
What would the value of the voltage
source have to be so it is equivalent to
the current source circuit?
VAB
Open
Short
RL
Vs
IAB
Vs
0
0
Vs
Rth
RL
RL  Rth
Vs
1
RL  Rth
What would the value of the current
source have to be so it is equivalent to
the voltage source circuit?
Kevin D. Donohue, University of Kentucky
6
Equivalent Circuits

Circuits containing different elements are
equivalent, if their response with respect to
(wrt) a pair of terminals is the same.

For the two previous circuits to be
equivalent, what would have to be true
about their source and resistance values?
Kevin D. Donohue, University of Kentucky
7
Source Transformation
The following circuit pairs are equivalent wrt to terminals AB.
Therefore, these source and resistor combinations can be swapped in a
circuit without affecting the voltages and currents in other parts of the
circuit

A
Is
Rth
B
Rth
A
Rth
Is Rth
B
A
Is
Vs
R th
A
Vs
Is
B
A
B
Kevin D. Donohue, University of Kentucky
Rth Vs
Rth
A
B
B
A
Vs
A
B
B
8
Source Transformation

Some equivalent circuits can be determined by
transforming source and resistor combinations and
combining parallel and serial elements around a terminal
of interest.

This method can work well for simple circuits with sourceresistor combinations as shown on the previous slide.

This method is limited, if dependent sources are present.
Kevin D. Donohue, University of Kentucky
9
Examples - Source Transformation

For several circuits find voltages and
currents in circuits with independent
sources and resistors using the method of
source transformation.
Kevin D. Donohue, University of Kentucky
10
Thévenin Equivalent Circuits

Find the value for Vth and Rth so the two circuits
will be equivalent at terminals AB.
Vs
Rth
+ VA -
A
A
2
B
2
12V
2
2 VA
+
Vo
-
B
Kevin D. Donohue, University of Kentucky
11
Norton Equivalent Circuits

Find the value for In and Rth so the two circuits
will be equivalent at terminals AB.
A
+ VA -
A
In
Rth
B
2
2
12V
2
2 VA
+
Vo
-
B
Kevin D. Donohue, University of Kentucky
12
Finding Thévenin and Norton Equivalent
Circuits

Identify terminal pair around which to find the
equivalent circuit.

Find voltage across the terminal pair when no load is
present (open-circuit voltage Voc)

Short the terminal and find the current in the short
(short-circuit current Isc)

Compute equivalent resistance as:
Rth = Voc / Isc
Kevin D. Donohue, University of Kentucky
13
Finding Thévenin and Norton Equivalent
Circuits

The equivalent circuits can then be
expressed in terms of these quantities
V
R th  oc
I sc
A
Isc
Rth
B
Voc
Rth
A
B
Kevin D. Donohue, University of Kentucky
14
Examples -Finding Equivalent Circuits

Find the Thévenin and Norton equivalents
for circuits containing independent and
dependent sources and resistors.

Show that for a maximum power transfer
from a circuit to a load resistor, it must
equal the Thévenin resistance of the circuit.
Kevin D. Donohue, University of Kentucky
15
SPICE Examples

Find the Thévenin and Norton equivalents
for circuits containing independent and
dependent sources and resistors.

Show that for a maximum power transfer
from a circuit to a load resistor, it must
equal the Thévenin resistance of the circuit.
Kevin D. Donohue, University of Kentucky
16